Question

Given that 1/Z = 1/Z1 + 1/Z2, obtain an explicit expression for R and X in terms of R1, R2, X1, X2.; where Z = R  + Xi,   Z1 = R1 +X1i, Z2 = R2 +X2i

Answer 1

This could take a little time, so I'll probably split the answer up into segments.

Z=Z1Z2/(Z1+Z2)=R+iX, so R is the real component and X the imaginary.

Z1Z2=(R1+X1i)(R2+X2i)=R1R2-X1X2+i(R1X2+R2X1)=A+iB where A=R1R2-X1X2 and B=R1X2+R2X1.

Z1+Z2=R1+R2+i(X1+X2)=C+iD where C and D are the real and imaginary parts.

So Z=(A+iB)/(C+iD)=(A+iB)(C-iD)/(C^2+D^2)=(AC+BD+i(BC-AD))/(C^2+D^2)=R+Xi.

Therefore, R=(AC+BD)/(C^2+D^2) and X=(BC-AD)/(C^2+D^2).

AC=(R1R2-X1X2)(R1+R2)=R1^2R2-R1X1X2+R1R2^2-R2X1X2.

BD=(R1X2+R2X1)(X1+X2)=R1X1X2+R2X1^2+R1X2^2+R2X1X2.

AC+BD=R1^2R2+R1R2^2+R2X1^2+R1X2^2.

C^2+D^2=(R1+R2)^2+(X1+X2)^2.

R=(R1^2R2+R1R2^2+R2X1^2+R1X2^2)/((R1+R2)^2+(X1+X2)^2).

BC=(R1X2+R2X1)(R1+R2)=R1^2X2+R1R2X1+R1R2X2+R2^2X1

AD=(R1R2-X1X2)(X1+X2)=R1R2X1-X1^2X2+R1R2X2-X1X2^2.

BC-AD=R1^2X2+R2^2X1+X1^2X2+X1X2^2.

X=(R1^2X2+R2^2X1+X1^2X2+X1X2^2)/((R1+R2)^2+(X1+X2)^2).